Role Assignments and Social Networks

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Role Assignments and Social Networks

• Fred S. Roberts
• Rutgers University
• Piscataway, NJ

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Role Assignments
Role assignments arise from the effort to model
the social roles that individuals play. The
motivating idea Individuals with the same role
will relate in the same way to other individuals
playing counterpart roles. Individuals occupying
the same position do not necessarily have similar
ties with the same other individuals, but they do
have the same ties with the same types of others.
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• Doctors have the same role-relations with
  patients, nurses, suppliers, and other doctors.
• They do not necessarily have the same
  role-relations with the same patients, nurses,
  etc.

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• Mothers do not have the same children. But they
  all have children.

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• The leader of a terrorist group does not
  necessarily relate to the fund-raiser for all
  other terrorist groups, but the leader relates to
  some terrorist group fund-raiser.

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Role assignments were formalized using concepts
of graph homomorphisms by Sailer (1978) and White
and Reitz (1983). We will follow the definition
of a role assignment (also called a role
coloring) given by Everett and Borgatti (1991).
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Social Network is represented by a graph G
(V,E). V individuals edge x,y means x, y
are related in some way. N(x) y x,y ? E
open neighborhood of x r(x) role assigned to
vertex x For simplicity r(x) ?
1,2,...,k. r(N(x)) r(y) y ? N(x)
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Role Assignment r(x) r(y) ? r(N(x))
r(N(y)) If two individuals have the same role,
they are related to individuals with the same
sets of roles.
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Example
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Indifference Graphs
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Indifference Graphs In role assignment model, we
place no significance on number defining role. We
don't ask if x has smaller role than y or if x
and y have roles that are close. A different
kind of model try to assign numbers to
individuals so that individuals who are related
are exactly the ones whose role-defining numbers
are close. The model Fix ? gt 0. x,y ? E ?
r(x) - r(y) lt ?. Graphs for which we can find
such an r are called indifference graphs.
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Example Indifference graphs have been
widely studied. They are easy to recognize. We
shall discuss the relation between the role
assignment and indifference graph models.
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k-Role Assignments
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k-Role Assignments Recall role assignment
means r(x) r(y) ? r(N(x)) r(N(y)). If r(V)
1,2,...,k, the role assignment r is called
a k-role assignment. G (V,E) is
k-role-assignable if it has a k-role
assignment.
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1-Role Assignable Graphs r(x) ? 1 is a role
assignment iff G has no isolated vertices or
all isolated vertices. V(G) -Role Assignable
Graphs r(x) all different is always a role
assignment, so every graph of n vertices is
n-role assignable.
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2-Role Assignable Graphs Given a k-role
assignment, build the corresponding role graph by
letting the vertices be 1,2,...,k and taking
an edge between i and j iff some vertex of
role i is adjacent to some vertex of role
j. If k 2, the possible role graphs
(unlabeled) are
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It is easy to check if G has a 2-role
assignment with role graph Ri , i 1, 2, 3, or
4. Let I set of isolated vertices in G. G
has a 2-role assignment with role graph R1 iff
I V(G). G has a 2-role assignment with role
graph R2 iff I ? V(G) and I ? ?.
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G has a 2-role assignment with role graph R3
iff I ? and G is disconnected. G has a
2-role assignment with role graph R4 iff I
? and G is bipartite.
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What about R5?
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Theorem (Roberts and Sheng) The problem of
determining if G has a 2-role assignment with
role graph R5 is NP-complete. (Proof is by
reducing 3-satisfiability to this problem.)
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Theorem (Roberts and Sheng) The problem of
determining if G has a 2-role assignment with
role graph R5 is NP-complete. (Proof is by
reducing 3-satisfiability to this problem.)
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Theorem (Roberts and Sheng) The problem of
determining if G has a 2-role assignment with
role graph R6 is NP-complete. Corollary The
problem of determining if G has a 2-role
assignment is NP-complete. Thus, there are
probably no good algorithms for determining
whether or not the 2-role model fits data.
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Role Assignments and Indifference Graphs
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Role Assignments and Indifference Graphs G
(V,E). Let x1 , x2 , ..., xn be an ordering of
V. We say the ordering is compatible if whenever
i ? j lt k ? l and xi,x1 ? E, then xj,xk ?
E. x1 , x2 , x3 , x4 , x5 , x6 is a
compatible ordering.
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Theorem (Roberts 1968) A graph G is an
indifference graph iff there is a compatible
order of vertices.
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Theorem (Roberts 1968) A graph G is an
indifference graph iff there is a compatible
order of vertices. Role Coloring an Indifference
Graph A Greedy Algorithm
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Let G be an indifference graph and x1 , x2 ,
..., xn be a compatible order. Let Nk(xk )
denote N(xk) ? xk ,xk1 , ..., xn. Greedy
Algorithm with 2 Roles r(xn ) ? 1 For i ? n-1 to
1 step -1, do if 1 ? r(Nk(xk)) or Nk (xk) 1
and xm ? Nk(xk ) and 2 ? r(Nm(xm)), r(xk) ?
2. otherwise r(xk) ? 1.
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Example
2
1
1
1
2
2
x1 ? 1 since 1 ? r(N1(x1)), N1(x1) 1,
x2 ? N1(x1), but 2 ? r(N2(x2)).
x6 ? 1
x5 ? 2 since 1 ? r(N5(x5))
x3 ? 1 since 1 ? r(N3(x3)) and N3(x3) gt 1
x4 ? 2 since 1 ? r(N4(x4))
x2 ? 2 since 1 ? r(N2(x2))
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Theorem (Sheng) If G is an indifference graph
with at most one vertex with only one neighbor,
then G has a 2-role assignment, obtainable by
using the greedy algorithm based on the
compatible ordering. If there are no isolated
vertices, the role graph is R5 . However, not
every graph with a compatible vertex ordering has
a 2-role assignment Example
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2-Role Assignable Indifference Graphs Simple
paths are 2-role assignable since they are
bipartite. Theorem (Sheng) Let G (V,E) be a
connected indifference graph with n gt 2 vertices
and two or more pendant vertices and assume that
G is not a simple path. Let x1 ,x2 ,...,xn be
a compatible order and let s,t ? 1,n be the
first and last i ? 1,n s.t. xi ,xi2 ? E.
Then
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(1). If s-1 ? 0 (mod 3) and n-t ? 0 (mod 3), G is
2-role assignable iff xs,xt ? E. (2). If s-1 ?
2 (mod 3) and n-t ? 0 (mod 3), G is 2-role
assignable iff xs,xt ? E or xs1 ,xt ?
E. (3). If s-1 ? 0 (mod 3) and n-t ? 2 (mod 3),
G is 2-role assignable iff xs,xt ? E or
xs,xt-1 ? E. (4). If s-1 ? 2 (mod 3) and n-t
? 2 (mod 3), and m ? s,t is the last i s.t.
xs ,xi ? E, and u ? s,t is the first i
s.t. xi,xt ? E, then G is 2-role assignable
iff u ? m or xs1,xt1 ? E. (5). If s-1 ?
1 (mod 3) or n-t ? 1 (mod 3), then G is always
2-role assignable.
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Consider previous example A compatible
order is given by x1 ,x2 ,..., x9. s 4, t
4 s-1 3 ? 0 (mod 3), n-t 5 ? 2 (mod
3) xs,xt ? E, xs,xt-1 ? E. Thus G is not
2-role assignable by part (3). Comment Sheng
has related results for triangulated graphs.
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Role Primitive Graphs
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Role Primitive Graphs G is role primitive if
the only role assignments are the trivial ones
where all vertices get the same role or all
vertices get different roles, and there are at
least 3 vertices. Theorem (Everett and Borgatti
(1991)) There is a role primitive
graph Question What is the smallest role
primitive graph?
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This graph is not an indifference graph it does
not have a compatible vertex ordering. Question
Are there role primitive indifference graphs?
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This graph is not an indifference graph it does
not have a compatible vertex ordering. Question
Are there role primitive indifference
graphs? Answer (Roberts and Sheng) Yes
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Theorem (Roberts and Sheng) Gp,q,w is not role
primitive if w ? 2. Gp,q,2 is an indifference
graph.
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Theorem (Roberts and Sheng) Gp,q,2 is not
2-role assignable iff p gt 0, q gt 0, p ? 0 (mod
3), and q ? 0 (mod 3). Theorem (Roberts and
Sheng) Let k ? 1,n-1 where n V(G). Then
Gp,q,2 is k-role assignable iff 1) p gt 0, q gt
0, and p,q ? k-2,k-1 or k-2,k (mod
2k-1) or 2) one arm has length 0 and the other
has length k-1 (mod 2k-1) or 3) p q k-2.
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Corollary G3,12,2 is a role primitive
indifference graph. It is not 2-role
assignable by the first theorem. It is not
k-role assignable for any k ? 1,n-1 by the
second theorem.
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Automorphisms of Role Primitive Graphs Theorem
(Everett and Borgatti 1991) If G is role
primitive, the only automorphism of G is the
identity. Sketch of Proof Lemma Let H be a
subgroup of Aut(G). Then the orbits of H form a
partition of V corresponding to the sets of
vertices of a given role in some role
assignment. Proof Suppose V1 , V2 , .., Vk
are the orbits and r(x) i if x ? Vi . Suppose
r(x) r(y). Therefore ? ? ? H s.t. ?(x) y. If
u ? N(x), then ?(u) ? N(?(x)), so ?(u) ? N(y).
But r(u) r(?(u)) by definition. Hence, r(N(x))
? r(N(y)). Similarly, r(N(y)) ? r(N(x)). Q.E.D.
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Proof of the Theorem By the Lemma, if G is
role primitive, either Aut(G) is the identity
or Aut(G) acts transitively. Suppose the latter.
By the lemma, the stabilizers must be trivial so
Aut(G) acts regularly. Since no subgroup of a
regular group can be transitive, Aut(G) cannot
contain subgroups. Thus, Aut(G) is of prime order
and therefore Abelian. But the only Abelian
automorphism groups which can act regularly on
the vertices of a graph are the elementary
Abelian 2-groups. Hence, Aut(G) Z2 ,
contradicting the fact that G has 3 or more
vertices. Q.E.D.
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The converse is false. Example
a 2-role assignment
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Question of Everett and Borgatti How common are
role-primitive graphs? Everett conjectured that,
asymptotically, almost all graphs are role
primitive. Pekec and Roberts showed that this
is in fact quite wrong Asymptotically, almost
all graphs are not role primitive. So While it
had been believed that for most social networks,
only trivial role assignments were feasible, this
shows the opposite.
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Variants on the Role Assignment Model

• Threshold Role Assignments

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Variants on the Role Assignment Model Threshold
Role Assignments If S and T are two sets of
numbers, let distance d(S,T) be defined
by d(S,T) mins-t s ? S, t ?
T. Convention d(?,?) 0, d(S,?) ? if S ?
?. Note not necessarily a metric d(S,T) can be
0 if S ? T also triangle inequality can fail.
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r is a threshold role assignment if r(x)
r(y) ? d(r(N(x)),r(N(y))) ? 1. If in addition
r(V) 1,2,...,k, we say it is a k-threshold
role assignment. Note that in contrast to role
assignments, the proximity of numbers
representing roles now means something.
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Example r(e) r(g) r(N(e)) 1,
r(N(g)) 2,3 d(1,2,3) 1.
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Theorem (Roberts) Every graph is k-threshold
role assignable for all k s.t. 2 ? k ? V(G).
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Theorem (Roberts) Every graph is k-threshold
role assignable for all k s.t. 2 ? k ? V(G).
Boring
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An Alternative Notion of Distance dH(S,T)
smallest p s.t. ?s ? S ? t ? T s.t. s-t ? p
and ? t ? T ? s ? S s.t. s-t ? p. Convention
dH(?,?) 0, dH (S,?) ? if S ? ?. dH is
called the Hausdorff distance. d(1,2,3,1)
0, dH(1,2,3,1) 2.
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r is a threshold close role assignment if
r(x) r(y) ? dH(r(N(x)),r(N(y))) ?
1. k-threshold close role assignment if in
addition r(V) 1,2,...,k
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Observe Every graph of at least 2 vertices is
2-threshold close role assignable. Why Use role
1 on all isolated vertices, role 2 on all others.
If no isolated vertices, use role 1 on one vertex
and role 2 on all others. The result follows
because dH(1,2,1) dH(1,2,2)
dH(1,2) 1.
Theorem (Roberts) Every graph of at least 3
vertices is 3-threshold close role assignable.
Theorem (Roberts and Sheng) Every graph of at
least 4 vertices is 4-threshold close role
assignable. Every graph of at least 5 vertices is
5-threshold close role assignable.
What about every graph of at least 6 vertices? k
vertices?
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Fitting the Role Assignment Model Approximately
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Fitting the Role Assignment Model
Approximately Rarely does a mathematical model
fit data perfectly. One is often satisfied if
the number of inconsistencies is negligible. To
make this precise, let r be a function from V
onto 1,2,...,k. To express how close r is to
a role assignment, we count the number of pairs
of vertices for which the requirement () r(x)
r(y) ? r(N(x)) r(N(y)) holds.
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Let Vi x r(x) i. Let Mi count the
fraction of all pairs of vertices x and y of
role i so that the condition for a role
assignment holds for x and y ()
r(N(x)) r(N(y)). Let A(x,y) 1 if () holds
and A(x,y) 0 otherwise. Let ni Vi .
Then
We take Mi 1 if Vi 1.
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Thus, one way to measure how close r is to a
k-role assignment is to use M(r) min Mi
. Taking the maximum over all possible r from
V onto 1,2,...,k gives a measure of how
close G is to being k-role assignable. We
make this precise by using ?k(G) max r
M(r). Maximum is over all assignments onto
1,2,...,k.
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A measure similar to ?k(G) arises in the theory
of blockmodeling in social network theory.
Here, we try to map a social network "almost
homomorphically" into a smaller network. General
goal replace a "large" network by a smaller one
fewer vertices that reflects its structural
relations. Implication Use as few roles as
possible.
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Example C5 To calculate ?2(G), we
consider all possible assignments of 1's and 2's
to the vertices with at least one of each.
Without loss of generality, we consider only the
cases where there are at most two 1's.
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Case 1 One 1 V1 x1, V2 x2 ,x3 ,x4 ,x5
. M1 1 since V1 1. The requirement ()
r(N(x)) r(N(y)) is satisfied in V2 only for
x2, x5 and x3, x4 so Thus, M(r) 1/3.
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Case 2 Two 1's on adjacent vertices. M1 1
since the requirement () r(N(x)) r(N(y))
holds for the one pair of vertices x1, x2 in
V1 . M2 1/3 the requirement holds for x3, x5
but fails for x3, x4 and x4, x5 . Thus, M(r)
1/3.
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Case 3 Two 1's not on adjacent vertices. Again,
M1 1 since the requirement holds on V1
x1, x3. On V2, the requirement holds for
x4, x5, but fails for x2, x4 and x2,x5. Thus,
M2 1/3. M(r) 1/3. Conclusion ?2(C5) 1/3.
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In fact C5 is the only graph with ?2 ?
1/3. The proof uses Theorem (Pekec and
Roberts) For every graph G of n ? 3
vertices n ? 0 (mod 2) ? ?2(G) ? 1/2 -
1/(2n-2) n ? 1 (mod 4) ? ?2(G) ? 1/2 -
1/(2n-4) n ? 3 (mod 4) ? ?2(G) ? 1/2
(n-5)/(2n2 -6n4)
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When is ?k(G) 1? Let ?(G) minimum degree of a
vertex in G (minimum number of neighbors). Theore
m (Pekec and Roberts) For k gt 1, if G is a
graph with n vertices and ?(G) gt
log(kn)/log(k(k-1)), then ?k(G) 1. Corollary
For all k gt 1, there is a constant ck such that
if ?(G) gt cklog n, then ?k(G)
1. Interpretation All social networks in which
each individual is involved in a significant
number of relationships can be captured by the
role assignment model with k roles.
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The proof is by the probabilistic method. It only
shows that such a k-role assignment exists. There
is no explicit construction that goes with
this. The proof of the theorem shows that,
asymptotically, almost all graphs are not role
primitive. This disproves a conjecture of Everett.
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Approximate Role Assignments with k2 Theorem
(Pekec and Roberts). Let G have n ? 3
vertices. Then ?2(G) gt 1 2 log2n
?/(n-2). Corollary For every ? gt 0, all but
finitely many graphs have ?2(G) gt 1- ?. Proof
For every ? gt 0, there is n? such that for all n
gt n? , 2 log2n /(n-2) lt ? . Therefore, all graphs
G of more than n? vertices have ?2(G) gt 1- ?.
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Approximate k-Role Assignments Theorem (Pekec
and Roberts). For all k gt 0, there is a
positive constant Ck such that for every graph
G on n ? k vertices, ?k(G) ? 1 - (Ck log
n)/n. Corollary. For every ? gt 0, there is n?
such that n gt n? implies that ?k(G) gt 1- ?. In
other words For any positive integer k, all but
finitely many graphs are "almost" k-role
assignable. That is, for all but finitely many
graphs, there is a k-role assignment such that
the fraction of pairs x,y satisfying the
condition for a role assignment is close to 1.
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Connections to Ecology
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Connections to Ecology Concepts of social network
theory, in particular the role assignment model,
have potentially useful applications in
ecology. Relevant areas of ecology Study of
community organization, food webs, and
biogeochemical cycles, with emphasis on network
structure. In the study of food webs,
graph-theoretic approaches similar to those used
to study social networks have been used to study
"trophic interactions." Borgatti, Everett,
Johnson, Luczkovich (2001) worked on defining and
measuring "trophic similarity" in food webs. They
found that the theory of role assignments is
relevant to the definition of "structural role"
in ecology.
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Open Problems
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Open Problems The theory of social networks is an
old one and it has given rise to many fascinating
graph-theoretical problems. Models of social
role lead to such problems. Here are a few of
the open questions that remain. 1. Characterize
or recognize 2-role assignable graphs, at least
under certain assumptions about the graphs. 2.
Investigate k-role assignable graphs for 2 lt k lt
V(G). Very little is known about these. 3.
Characterize role primitive graphs.
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4. What is the smallest role primitive graph? 5.
Develop methods for determining if a graph has a
k-threshold close role assignment. 6. Is every
graph of at least k vertices k-threshold role
assignable? 7. Find constructions of graphs G
for which ?k(G) is large.
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Thank you!